Past Advanced Class Logic

Elekes and Szabó established non-trivial incidence bounds for binary algebraic relations in characteristic 0, generalizing the Szemerédi-Trotter theorem for point-line-incidence. This was later generalized to binary relations defined in reducts of so-called distal structures in a result of Chernikov, Peterzil and Starchenko. For fields of positive characteristic, such bounds fail to hold in general. Bays and Martin apply the bounds for distal structures in the context of valued fields to derive incidence bounds in the sense of Szemerédi-Trotter in fields admitting valuations with finite residue field, such as F_p(t). We show that this result can be made uniform in the size of the finite residue field, by making precise in some sense the intuition that ACVF is distal relative to the residue field. In this talk, I will introduce the relevant notions from incidence combinatorics and distality, before outlining a proof of the uniform-in-p result.

The theory of characters for infinite groups, initiated by Thoma, is a natural generalization of the representation theory of finite groups. More precisely, a character on a discrete group is a normalised positive definite function which is conjugation invariant and extremal. Connes conjectured a rigidity result for characters of an important family of discrete groups, namely, irreducible lattices in higher-rank semisimple Lie groups. The conjecture states that every character is either the trace of a finite-dimensional representation, or vanishes off the center. This rigidity property implies the Stuck-Zimmer conjecture for such lattices, namely, ergodic actions are either essentially transitive or essentially free. I will present a recent joint result with Michael Glasner, Yuval Gorfine, Liam Hanany and Arie Levit in which we prove that non-uniform irreducible lattices in higher-rank semisimple groups are character rigid. As a result, we also obtain a resolution of the Stuck-Zimmer conjecture for all non-uniform lattices.

This talk focuses on the logic side of the following result: the non-definability of free independence in the theory of tracial von Neumann algebras and C*-probability spaces. I will introduce continuous model theory, which is suitable for the study of metric structures. Definability in the continuous setting differs slightly from that in the discrete case. I will introduce its definition, give examples of definable sets, and prove an equivalent ultrapower condition of it. A. Berenstein and C. W. Henson exposited model theory for probability spaces in 2023, which was done with continuous model theory. It makes it natural for us to consider the definability of the notion of free independence in probability spaces. I will explain our result, which gives an example of a non-definable set.

This is work with William Boulanger and Emma Harvey, supervised by Jenny Pi and Jakub Curda.

I will talk about some recent work with Tingxiang Zou on higher-dimensional Elekes-Szabó problems in the case of an Ind-constructible action of a group G on a variety X. We expect nilpotent algebraic subgroups N of G to be responsible for any such; this roughly means that if H and A are finite subsets with non-expansion |H*A| <= |A|^{1+\eta}, then H concentrates on a coset of some such N.

A natural example is the action of the Cremona group of birational transformations of the plane. I will talk about a recent result which confirms the above expectation when we restrict to the group of polynomial automorphisms of the plane, using Jung's description of this group as an amalgamated free product, as well as some work in progress which combines weak polynomial Freiman-Ruzsa with effective Mordell-Lang, after Akshat Mudgal, to handle some further special cases.

Building on Leo’s talk last week, I will present the full Galois characterisation of henselianity and introduce some of the ‘explicit’ ingredients he referred to during his presentation. In particular, I will describe a Galois cohomology-inspired criterion for distinguishing between different characteristics. I will then outline the full proof of the Galois characterisation of p-adically closed fields, indicating how each of the ingredients enters the argument.

The absolute Galois group of ℚₚ determines its field structure: a field K is p-adically closed if and only if its absolute Galois group is isomorphic to that of ℚₚ. This Galois-theoretic characterisation was proved by Koenigsmann in 1995, building on previous work by Arason, Elman, Jacob, Ware, and Pop. Similar results were obtained by Efrat and further developed in his 2006 book.

Our project aims to provide an optimal proof of this characterisation, incorporating improvements and new developments. These include a revised proof strategy; Efrat's construction of valuations via multiplicative stratification; the Galois characterisation of henselianity; systematic use of the standard decomposition; and the function field analogy of Krasner-Kazhdan-Deligne type. Moreover, we replace arguments that use Galois cohomology with elementary ones.

In this talk, I will focus on two key components of the proof: the construction of valuations from rigid elements, and the role of the function field analogy as developed via the non-standard methods of Jahnke-Kartas.

This is joint work with Jochen Koenigsmann and Benedikt Stock.

I introduce modal group theory, where one investigates the class of all groups using embeddability as a modal operator. By employing HNN extensions, I demonstrate that the modal language of groups is more expressive than the first-order language of groups. Furthermore, I establish that the theory of true arithmetic, viewed as sets of Gödel numbers, is computably isomorphic to the modal theory of finitely presented groups. Finally, I resolve an open question posed by Sören Berger, Alexander Block, and Benedikt Löwe by proving that the propositional modal validities of groups constitute precisely the modal logic S4.2.

We establish that a criterion based on ring-theoretic amenability is both necessary and sufficient for the abelian version of the Elekes-Szabó theorem to be sharp in the case of positive characteristic. Moreover, the criterion is always sufficient. We provide illustrative examples in the theories ACF_p and DCF_0.

I will talk about intersection theory over any globally valued field and how it is connected to some model-theoretic problems.

We shall explain how to represent a couple of basic notions in model theory by standard simplicial diagrams from homotopy theory. Namely, we shall see that the notions of a {definable/invariant type}, {convergence}, and {contractibility} are defined by the same simplicial formula, and so are that of a {complete E-M type} and an {idempotent of an oo-category}.  The first reformulation makes precise Hrushovski's point of view that a definable/invariant type is an operation on types rather than a property of a type depending on the choice of a model, and suggests a notion of a type over a {space} of parameters. The second involves the nerve of the category with a single idempotent non-identity morphism, and leads to a reformulation of {non-dividing} somewhat similar to that of lifting idempotents in an oo-category. If time permits, I shall also present simplicial reformulations of distality, NIP, and simplicity.

We do so by associating with a theory the simplicial set of its n-types, n>0. This simplicial set, or rather its symmetrisation, appeared earlier in model theory under the names of {type structure}  (M.Morley. Applications of topology to Lw1w. 1974), {type category} (R.Knight, Topological Spaces and Scattered Theories. 2007), {type space functors} (Haykazyan. Spaces of Types in Positive Model Theory. 2019; M.Kamsma. Type space functors and interpretations in positive logic. 2022).

I will talk about preliminaries in Arakelov geometry. Also, a historical overview will be provided. This talk will be the basis of a later talk about the theory of globally valued fields.

In algebraic geometry, the technique of dévissage reduces many questions to the case of curves. In difference and differential algebra, this is not the case, but the obstructions can be closely analysed. In difference algebra, they are difference varieties defined by equations of the form \si(𝑥)=𝑔𝑥\si(x)=gx, determined by an action of an algebraic group and an element g of this group. This is joint work with Zoé Chatzidakis.

In algebraic geometry, the technique of dévissage reduces many questions to the case of curves. In difference and differential algebra, this is not the case, but the obstructions can be closely analysed. In difference algebra, they are difference varieties defined by equations of the form $\si(x)=g x$, determined by an action of an algebraic group and an element g of this group. This is joint work with Zoé Chatzidakis.

I will talk about the connections between the Siu inequality and existence of the model companion for GVFs. The talk will be partially based on a joint work with Antoine Sedillot.

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of Q_p^ab, the maximal extension of the p-adic numbers Q_p with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of Q_p) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

Jochen Koenigsmann’s Habilitation introduced a classification of fields elementarily characterised by their absolute Galois groups, including two conjecturally empty families. The emptiness of one of these families would follow from a Galois cohomological conjecture concerning radically closed fields formulated by Koenigsmann. A promising approach to resolving this conjecture involves the use of local-global principles in Galois cohomology. This talk examines the conceptual foundations of this method, highlights its relevance to Koenigsmann’s classification, and evaluates existing local-global principles with regard to their applicability to this conjecture.

Please attend if you would like to give a talk in the Logic Advanced Class this term.

Solovay defined the inner model $L(\mathbb{R}, \mu)$ in the context of $\mathsf{AD}_{\mathbb{R}}$ by using it to define the supercompactness measure $\mu$ on $\mathcal{P}_{\omega_1}(\mathbb{R})$ naturally given by $\mathsf{AD}_{\mathbb{R}}$. Solovay speculated that stronger versions of this inner model should exist, corresponding to stronger versions of the measure $\mu$. Woodin, in his unpublished work, defined $\mu_{\infty}$ which is arguably the ultimate version of the supercompactness measure $\mu$ that Solovay had defined. I will talk about $\mu_{\infty}$ in the context of $\mathsf{AD}^+$ and the axiom $\mathsf{V} = \mathsf{Ultimate\ L}$.

https://woloszyn.org/

Does the limit construction for inverse systems of first-order structures preserve elementary equivalence? I will give sufficient conditions for when this is the case. Using Karp's theorem, we explain the connection between a syntactic and formal-semantic approach to inverse limits of structures. We use this to give a simple proof of van den Dries' AKE theorem (in ZFC), a general AKE theorem for mixed characteristic henselian valued fields with no assumptions on ramification. We also recall a seemingly forgotten result of Feferman, that can be interpreted as a "saturated" AKE theorem in positive characteristic: given two elementarily equivalent $\aleph_1$-saturated fields $k$ and $k'$, the formal power series rings $k[[t]]$ and $k'[[t]]$ are elementarily equivalent as well. We thus hope to popularise some ideas from categorical logic.

Consider definable sets over the family of finite fields $\mathbb{F}_q$. Ax proved a quantifier-elimination result for this theory, in a reasonable geometric language. Chatzidakis, Van den Dries and Macintyre showed that to a first-order approximation, the cardinality of a definable set $X$ is definable in a very mild expansion of Ax's theory.  Can such a statement be true of the next higher order approximation, i.e. can we write $|X(\mathbb{F}_q)| = aq^{d} + bq^{d-1/2} + o(q^{d-1/2})$, with $d,a,b$ varying definably with $X$ in a tame theory?    Here $b$ must be viewed as real-valued so continuous logic is needed. I will report on joint work in progress with Will Johnson.

I will report on a joint work with Pablo Destic and Nuno Hultberg, about some applications of Globally Valued Fields (GVFs) and I will describe a density result that we needed, which turns out to be connected to Riemann-Zariski and Berkovich spaces.

I will present a recent work with G. Kocharyan, where we show the undecidability of the following two problems: given a finitely generated subgroup G of GL(n,Q), a) determine whether G has a non-identity element whose (i,j) entry is equal to zero, and b) determine whether the stabilizer of a given vector in G is non-trivial. Undecidability of problem b) answers a question of Dixon from 1985. The proofs reduce to the undecidability of the word problem for finitely presented groups.

Motivated by work of Hrushovski on pseudofinite fields with an additive character we investigate the theory ACFA+ which is the model companion of the theory of difference fields with an additive character on the fixed field. Building on results by Hrushovski we can recover it as the characteristic 0-asymptotic theory of the algebraic closure of finite fields with the Frobenius-automorphism and the standard character on the fixed field. We characterise 3-amalgamation in ACFA+. As cosequences we obtain that ACFA+ is a simple theory, an explicit description of the connected component of the Kim-Pillay group and (weak) elimination of imaginaries. If time permits we present some results on higher amalgamation.

We will decide on speakers for Trinity term 2024.

I will talk about some model-theoretic properties of Booleanizations of theories, subdirect products of structures, and sheaves of structures. I will discuss a result of Macintyre from 1973 on model-completeness, and more recent results jointly with Ehud Hrushovski and with Angus Macintyre.

I will discuss aspects of some work in progress with Tingxiang Zou, in which we continue the investigation of pseudofinite sets coarsely respecting structures of algebraic geometry, focusing on algebraic group actions. Using a version of Balog-Szemerédi-Gowers-Tao for group actions, we find quite weak hypotheses which rule out non-abelian group actions, and we are applying this to obtain new Elekes-Szabó results in which the general position hypothesis is fully weakened in one co-ordinate.

This talk is based on a joint work with Vincent Jinhe Ye. I will define various classes of hyperbolic varieties (Broody hyperbolic, algebraically hyperbolic, bounded, groupless) and discuss existence of model companions of classes of fields that exclude them. This is related to moduli spaces of maps to hyperbolic varieties and to the (open) question whether the above mentioned hyperbolicity notions are in fact equivalent.

Let q be a prime power and let C be a smooth curve defined over F_q. The number of points of C over the finite extensions of F_q are determined by the Zeta function of C, which can be written in the form P_C(t)/((1-t)(1-qt)), where P_C(t) is a polynomial of degree 2g and g is the genus of C; this is often called the L-polynomial of C. We use a Chebotarev-like statement (over function fields instead of Z) due to Katz in order to study the distribution, as C varies, of the coefficients of P_C(t) in a non-archimedean setting.

This is a pre-seminar meeting for Margaret Bilu's talk "A motivic circle method", which takes place later in the day at 5PM in L3.

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of Q_p^ab, the maximal extension of the p-adic numbers Q_p with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of Q_p) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method can yield short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

For a complete theory T, Lascar associated with it a Galois group which we call the Lacsar group. We will talk about some of my work on recovering the Lascar group as the fundamental group of Mod(T) and some recent progress in understanding the higher homotopy groups.

The local Kronecker-Weber theorem states that the maximal abelian extension of p-adic numbers Q_p is obtained from this field by adjoining all roots of unity. In 2018, Koenigsmann conjectured that the maximal abelian extension of Q_p is decidable. In my talk, we will discuss Koenigsmann's proposed axiomatisation. In contrast, the maximal unramified extension of Q_p is known to be decidable, admitting a complete axiomatisation by an informed but simple set of axioms (this is due to Kochen). We explain how the question of completeness can be reduced to an Ax-Kochen-Ershov result in residue characteristic 0 by the method of coarsening.

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).

I will give an overview of new ideas showing up in arithmetic intersection theory based on some exciting talks that appeared at the very recent conference "Global invariants of arithmetic varieties". I will also outline connections to globally valued fields and some classical problems.

.. showing that a field K is isomorphic to Q if it has the same absolute Galois group and if it satisfies a very small additional condition (very similar to my talk 2 years ago).

What makes an intersection likely or unlikely? A simple dimension count shows that two varieties of dimension r and s are non "likely" to intersect if r < codim s, unless there are some special geometrical relations among them. A series of conjectures due to Bombieri-Masser-Zannier, Zilber and Pink rely on this philosophy. I will speak about a joint work with F. Barroero (Basel) in this framework in the special case of a curve in a family of elliptic curves. The proof is based on Pila-Zannier method, combining diophantine ingredients with a refinement of a theorem of Pila and Wilkie about counting rational points in sets definable in o-minimal structures.
   Everyone welcome!
 

Using results by Eisenbud, Schoutens and Zilber I will propose a model theoretic structure that aims to capture the algebra (or geometry) of a non reduced scheme over an algebraically closed field.